volume 58 article #2

Annual Effective Interest Rates When the Term Structure Is Stochastic

Jeffrey R. Stokes

The author is an assistant professor in the Department of Agricultural Economics and Rural Sociology, Pennsylvania State University, University Park.

Abstract

For truth-in-lending type compliance, lenders must disclose information about loans to borrowers, including the annual percentage rate of interest. Inconsistency and confusion over which noninterest costs to include, the failure to account for compounding, and the typical implicit assumption of a deterministic and flat term structure of interest rates imply the annual percentage rate disclosed by lenders is potentially meaningless. This is especially true for adjustable-rate mortgages. Monte Carlo simulations of the annual effective rate of interest under popular parametric term structure models reveal, in most cases, that the deterministic annual percentage rate of interest understates the true cost of financing when interest rates are relatively low or near their long-term average, but may overstate the true cost of financing when interest rates are relatively high.

Key words: adjustable-rate mortgage, annual percentage rate, annual effective rate, truth in lending, Monte Carlo simulation.

Article <top>

Compliance with the Truth-in-Lending Act (TILA) forces lenders to disclose detailed information about the products they offer. The Farm Credit Act (FCA) of 1971, as amended by the Farm Credit Amendments Act of 1985 and the Agricultural Credit Act of 1987, extended consumer protection to Farm Credit System loans. FCA Regulations 12 CFR, Part 614, Subpart K, requires qualified lenders to provide truth-in-lending type disclosures on loans exempt from the TILA. With respect to real estate mortgages in general, recent amendments to the TILA, like the Home Ownership Equity Protection Act of 1994, have imposed additional disclosure requirements and substantive limitations on certain closed-end mortgage loans with rates or fees above certain percentages or amounts (Financial Update staff, p. 3). Thus the vast majority of (if not all) real estate mortgages are subject to some form of truth-in-lending type regulation and disclosure.

More specifically, Farm Credit System disclosure regulations dictate that the following items must be provided in writing to a prospective borrower not later than the time of loan closing:

• A reporting of the current rate of interest on a loan;

• The amount and frequency of interest rate adjustments during the term of adjustable-rate loans, as well as specific standard adjustment factors affecting the interest rate adjustment, or if there are no limitations on the amount and frequency of the adjustments, a statement to that effect;

• The current effective interest rate, taking into account stock purchases or participation certificate purchases and any loan origination charges, with one or more representative examples of the impact of these amounts on the stated interest rate;

• A statement that the stock is at risk (except for eligible borrower stock under § 4.9A of the Act); and

• A statement indicating the various types of loan options available to borrowers with an explanation of the terms and borrower rights that apply to each type of loan.

There are at least three problems associated with the above disclosure guidelines. First, most lending institutions disclose an annual percentage rate (APR) of interest rather than the annual effective rate (AER) of interest. While not overly crucial (because the AER can be inferred from the APR), the APR does ignore the effects of compounding and typically will understate the cost of credit, thus providing a somewhat misleading figure. Second, the banking community in general has yet to form consensus regarding which noninterest costs are relevant for calculating the APR. This fact is apparent by the lack of detail provided in the above listing for Farm Credit Service loans. Last, adjustable rate mortgages (ARMs) have become a very popular product over the last few years, and the method used to calculate the APR does not take into account the fact that interest rates are stochastic. A stochastic term structure will alter the future repayment stream needed to retire the loan, and therefore affects the calculation of the APR.

Given these limitations, the purpose of this study is threefold. First, I briefly argue in favor of the AER over the APR while stressing the importance of including all the relevant costs associated with a financing source for more accurately assessing the cost of credit. Second, I present an extension to a well-known methodology for determining the AER that accounts for a stochastic term structure, thereby providing a means of examining the cost of credit for adjustable-rate mortgages. Finally, the empirical distributions of the AER under assumed loan parameters and term structures are examined in detail and inferences are drawn. More accurate cost of credit estimation would prove beneficial to all consumers when shopping for credit. Agricultural producers could further benefit from more accurate disclosure if the cost of credit is a needed component for planning purposes. Examples include capital budgeting and financial analysis.

A Simple Actuarial Interest Rate Formula <top>

Given information about the terms of a loan and the noninterest costs associated with using funds from a particular source, the APR can be determined precisely in the case of a fixed-rate mortgage₁. Specific information is needed concerning the terms of the loan: the contractual or stated rate of interest, the frequency of repayment, the length of the loan, interest calculation method, and type of repayments. Numerous approximating methods are used in practice; however, the following method appears to offer the greatest potential owing to its intuitive nature as well as its ease of implementation.

¹Typically, the only noninterest costs that are always included in the APR calculation are points and per diem interest. As noted above, Farm Credit Service loans also include stock purchases or participation certificate purchases and any loan origination charges. Those costs that are sometimes included, depending on the lender, are: application fees, processing completion fees, underwriting fees, private mortgage insurance, and credit life insurance. More likely to not be included in the APR calculation are closing costs (including transfer taxes, title examination and abstract fees, property surveys, attorney fees, notary fees, document preparation fees, and recording fees), engineering costs, and pest-control fees. Also typically not included are any balloon payments and prepayment penalties for retiring a loan early.

The formula is given as:

In equation (1), C₀ represents the loan proceeds (i.e., the amount borrowed), P_t is the time t payment that includes noninterest costs, T is the total number of payments required to retire the loan, and i_act is the actuarial rate of interest. Equation (1) is relatively simple to implement in practice because the equation is very much like a net present value equation where C₀ is analogous to an initial investment, the P_t are analogous to period t net cash flows, and i_act is analogous to a discount rate.

Actual implementation of equation (1) proceeds in the following manner. Noninterest costs are determined and added to the loan amount. Given a contractual interest rate and other terms of the loan, this amount is amortized over the appropriate number of repayment periods. The resulting P_t and C₀ are then used in equation (1) to determine the actuarial rate of interest. In this sense, equation (1) is used in a manner similar to an internal rate of return calculation.

The actuarial rate of interest is the rate of interest per conversion period and includes interest and noninterest costs. From an estimate of the actuarial rate of interest, the annual percentage and annual effective rates of interest can be determined by applying the following two formulas:

where i_apr is the annual percentage rate of interest, i_eff is the annual effective rate of interest, and m is the number of conversion periods per year.

Conceptually, then, the actuarial rate of interest is the rate per conversion period that would make a risk-neutral borrower indifferent between paying the contractual interest rate on the loan and noninterest costs, and paying the APR implied by the actuarial interest rate and no noninterest costs. Equating the loan amount with the amortization of this loan amount including noninterest costs in the manner suggested allows for the explicit consideration of the impact of noninterest costs by embedding their effect into the contractual interest rate.

At least three potential problems exist with the simple interest rate model implied by formulas (1) and (2) as they relate to truth-in-lending type disclosure. As noted above, there is little consensus regarding which costs to include in the APR calculation and, in addition, the APR is inferior to the AER for accurately assessing the cost of credit. From an economic perspective, it would be difficult to justify excluding any noninterest costs associated with a particular financing source. This is especially true when trying to compare the costs of financing from two different sources offering different contractual rates of interest.

Even when two competing lenders offer the same contractual interest rate, differences in total noninterest costs and/or the specific terms of the loan can result in differences in the APR. For example, the APR is higher for constant payment on principal loans when compared to constant payment loans, while the APR increases as the frequency of repayments increases but decreases as the loan amount and length of loan increase. Additionally, through reliance on the actuarial rate, the AER provides the same informational content as the APR but also offers the advantage of explicitly accounting for the effect of compounding.

A third, and potentially more critical problem for adjustable rate mortgages is that the model is deterministic. On the one hand, if the loan is a fixed-rate loan, no real problems are created because the rate disclosed will be entirely accurate for this type of product assuming all costs are included as noted above. However, adjustable-rate mortgages are at least as common, and equation (1) does not allow for the impact of stochastic interest rates for these types of mortgages₂. Adjustable-rate mortgages, sometimes referred to as variable- or floating-rate mortgages, are loans where the interest rate changes periodically in relation to an index. One-, three-, and five-year Arms are quite common, and typically the annualized yield on a U.S. Treasury security is used as the index. If interest rates are stochastic, the future payments to retire the loan (after one, three, or five years, for example) are unknown at the inception of the loan. Thus, the APR and AER are also unknown, and it is not entirely clear why they should be very accurately identified by the deterministic approach outlined above and commonly used in practice.

²One could argue the model does hold for stochastic interest rates that always equal their expected value in subsequent periods. Such a model of the term structure is not, however, very realistic or enlightening.

An Alternative Actuarial Interest Rate Approximation <top>

Monte Carlo simulation can be used to enumerate a variant of equation (1) by assuming a particular term structure for the underlying interest rate (index), simulating "many" outcomes of equation (1) under alternative interest rate scenarios, and averaging over these outcomes. It should be noted that adjustments to simulated yields on US Treasury securities typically need to take place to account for any markup the lender may charge over the yield, and intra- and inter-year rate caps and/or floors. Given a particular term structure, then, it is not discernable which probability distribution AERs follow because the distribution is most likely truncated on one or both tails. Nevertheless, the empirical distribution can be examined in detail.

Numerous term structure models are at the disposal of the practitioner and applied modeler. The vast majority of these models are continuous-time, stochastic differential equation specifications because the models are most typically used to price derivative securities. Such one-factor models often offer the analytic convenience needed to solve partial differential equations describing the value of derivatives depending fundamentally on the term structure₃. Restricting attention to these parametric diffusion models, a general specification can be stated as

where r_t is a real-time continuous stochastic process for the short-term interest rate, t is an index for time,  and are generally specified drift and diffusion terms, and dZt is the differential of a Weiner process with E(dZt) = dt.  Given these definitions, can be interpreted as the expected rate of change in the short-term interest rate, while is the standard deviation of the changes in the short-term rate.

At least nine different term structure models are commonly used in empirical work. In these models, the functional form of the drift and diffusion terms are explicitly specified. Where applicable, the authors who first used a particular model, date used, and functional form of the model are listed in Table 1.₄
³One-factor models are generally defined as being dependent on only one state variable.
⁴More detailed information about the properties of these term structure models and their use can be found in Chan, Karolyi, Longstaff, and Sanders, or in the actual research papers listed in the reference section.

Table 1. Term Structure Models and Functional Forms

^a The Unrestricted, Geometric Brownian Motion, and Constant Elasticity of Variance models are not directly attributable to any one researcher or research.

As noted by an anonymous reviewer, the models presented in Table 1 are all equilibrium-based models as opposed to arbitrage-free models (Hull). While market practitioners are more likely to use an arbitrage-free model of the short-term rate to mark their portfolios to market, the equilibrium models presented in Table 1 are, however, all Heath-Jarrow-Morton (HJM) consistent models. The implication of HJM is there is a mathematical transformation that makes the two alternative descriptions equivalent, albeit under a change of probability measure. Such a change in measure is accomplished through the application of the Cameron-Martin-Girsanov theorem and may induce drastic (or very subtle) changes in the drift and diffusion terms of the stochastic differential equations listed in Table 1. Such transformations are well beyond the scope of the present research.

Given estimates of the parameters for each term structure model, the stochastic differential equations listed in Table 1 can be simulated. With more specific information about the terms of a loan, the markup (above the simulated yield) the lender charges, and other specific information like interest rate caps and/or floors, the prevailing rate during the adjustment periods can be determined. Subsequently, the payments associated with the rate can be determined. Once the payment series for the entire length of the loan is computed, equation (1) can then be applied to find the actuarial rate from which the APR and AER can be determined.

Replicating this experiment "many" times and averaging across replications will potentially give a more accurate estimate of the "true" cost of financing because the stochastic nature of interest rates will have been taken into account. At least four problems still remain. First, how many replications for a given term structure are necessary? Based on the Central Limit Theorem, the following can be stated (Campbell, Lo, and MacKinlay):

where R is the number of replications, î_act is the Monte Carlo estimate of the actuarial rate of interest, and i_act is the true actuarial rate of interest. Given the assumptions on the diffusion equations presented in Table 1, the Central Limit Theorem can be applied to approximate the number of replications. The distribution for the expression in equation (4) is normal with zero mean and variance related to n, the number of discrete intervals in the [0, T] time interval. Confidence intervals for i_act can be determined for a given level of confidence, and the approximate number of replications can be inferred.

A second problem, however, is that there is no guarantee the term structure model chosen is the "true" term structure. As noted by Fischer and Pederson, the evaluation of ARM loans is critically dependent on the specification of the stochastic behavior of interest rates. Consequently, an estimate of the AER under the wrong term structure model for the index may be inaccurate. This may be especially problematic if recent results reported by Stanton are valid—i.e., the author found nonlinearity in US Treasury security data. Stanton’s results indicate there is intense downward pressure on the drift term and intense upward pressure on the diffusion term in equation (1) when the short-term rate increases beyond about 14%. Such mean-reverting behavior is captured by several, but not all, of the models presented in Table 1 (those where α and β exist allow for mean-reverting behavior). If Stanton’s findings hold, it is likely that a model precluding mean version over some portion of the domain of short rates while allowing mean reversion over other portions of the domain could outperform the models listed in Table 1.

A third remaining problem is that borrowers can refinance adjustable-rate mortgages when it is desirable to do so. Falling interest rates which appear to be at a local minimum frequently induce many borrowers to "buy down" their interest rate and "lock-in" the resulting lower rate for the remainder of the loan. Such a phenomenon is not captured in the present approach, but most assuredly affects the actuarial rate of interest because the future payment series is altered (held constant at the "locked-in" rate).

Finally, early repayment (or the probability of early repayment) and associated penalties are not easy to assess at the inception of a loan, but do affect the repayment stream; as such, they have implications for the cost of credit. Such a phenomenon also puts the lender at risk. One of the driving forces behind the risk of prepayment is relocation, which is generally more of an issue for home mortgage lending than agricultural real estate mortgages. Nonetheless, prepayment is still a reality for agricultural loans, as there are other reasons (besides relocation) for early repayment.

An Empirical Example <top>

In this section, deterministic AERs are compared with the stochastic AERs estimated via Monte Carlo simulation under each of the term structure models presented in Table 1. Although hypothetical loan parameters are used, they are typical of contemporary mortgages offered by commercial banks or Farm Credit Services.

The specific loan assumed is a real estate mortgage for a $150,000 parcel of land. A 20% down payment is assumed to be required, making the loan amount $120,000 ($30,000 down payment). The borrower agrees to a one-year ARM with adjustments based on the monthly average yield on US Treasury securities adjusted to a constant maturity of one year, as made available by the Federal Reserve Board, plus a 200 basis point markup. Repayments are made monthly over a 20-year period, and the ARM is assumed to have a period (monthly) adjustment cap of 200 basis points with a maximum rate over the life of the loan of 12%. Noninterest costs are assumed to be $3,000, and include a $1,000 stock purchase similar to that required by Farm Credit Services. The stock purchase is assumed to be refunded when the loan is fully repaid (i.e., after the 240th and final payment). The balance of the noninterest costs are unspecified, but are assumed to stem from the list presented earlier in footnote 1. More explicitly, specified noninterest costs also could have been included with a more exact dollar figure, but the present assumption of $3,000 is adequate for demonstration purposes.

Parameter estimates for the term structure models are presented in Table 2. The estimates are from Chan et al., who use Hansen’s Generalized Method of Moments to estimate the parameters using discretely sampled data. The data are annualized, one-month US Treasury bill yields over the period June 1964 through December 1989, resulting in 306 observations. Nowman’s Gaussian parameter estimates also could have been used, but the temporal aggregation bias induced by the Chan et al. approach is minimal, making the exact advantage of using Nowman’s estimates indeterminate. These data are not necessarily reflective of current market conditions and as such, their usefulness in absolute terms is questionable. A more up-to-date analysis would entail applying the methods of Chan et al. to newer data. Thus, while not to be taken literally, the results still can be generalized for demonstration purposes. Also reported in Table 2 are the estimated expected long-term mean yields (η) for those models allowing mean-reverting behavior as discussed above₅.

⁵ Using the unrestricted model as an example, mean-reverting models imply that the long-term mean interest rate to which the short-term rate tends to revert is given by ­α/β, and the speed with which the reversion occurs is given by ­β.

Presented in Table 3 are the deterministic AER estimates under three different contractual interest rates found by applying equations (1) and (2). Further, Table 3 provides summary statistics for the AERs associated with Monte Carlo simulations assuming a one-year ARM under the same contractual interest rates and starting values₆. Data in the rightmost column come from the empirical distribution, and are the estimated probabilities of achieving a stochastic AER greater than the deterministic AER. Although not reported, the variance estimates of the AER simulations indicate anywhere from about 150 to 2,000 replications are needed to construct a 95% confidence interval that the AER estimate is within 10 basis points of the true value, depending on the term structure in question. The unrestricted term structure model has the smallest variance and requires the least number of replications, while the Merton term structure has the largest variance and requires the most. Recall that the number of replications is approximate, owing to the short-term rate caps and floors and their impact on the distribution of actuarial rates of interest.

⁶ More specifically, the contractual rate of interest is used in the determination of the first 12 payments. After this time, and each year subsequent to this time, the rate of interest used to determine payments over the next year will be exactly 2% above the stochastic one-month US Treasury securities annualized yield. Exceptions are when rates rise above or fall below 2% of the previous year’s rate, in which case the rate is capped or floored by this amount. Additionally, the rate cannot rise above 12%, and is therefore capped at this level as well. While at least one term structure model, that of Merton (1973), does not preclude negative interest rates, the model used in the analysis floors interest rates at 0.0%.

As shown in Table 3, contractual rates of interest of 6% and 8% generally produce simulated AERs in excess of those estimated using the deterministic method. Exceptions are the term structure model implied by Dothan and that implied by CIR-VR. The fact that both of these term structure models behave similarly is not surprising inasmuch as the Dothan model and the CIR-VR model are very closely related (see Table 1). The reason these two models predict annual AERs that are less than their deterministic counterparts can be traced to the restriction that the drift term is zero. All the other models have positive drift. Even more importantly, both the Dothan term structure model and the CIR-VR term structure model generally predict AERs that are less than the contractual interest rates.

Simulated AERs are higher than the deterministic AERs because the term structure models in question put upward pressure on interest rates, or at least a positive probability that rates will increase. This implies at least some of the future payments to retire the loan will be higher because the interest paid is higher. For those models for which a long-term mean is present, the 6% contractual rate of interest is below the long-term mean in all cases. Additionally, two of the models predict long-term rates in excess of the 8% contractual rate as well. Given this, upward pressure on interest rates occurs.

When the contractual rate of interest is 10%, the preceding results do not generally hold. In fact, only the Merton model and the GBM model predict AERs higher than those predicted by the deterministic model. Most likely for some of the models, the cause is again the long-term rate implied by the term structure in that the 10% contractual rate is higher than any of the long-term rates. Thus, there is downward pressure on interest rates.

Given these generalizations, it is not surprising that in the majority of the term structure models, the probability of observing a simulated AER in excess of the deterministic rate is very high for low to moderate contractual rates of interest. For example, in all cases except the Dothan and CIR-VR term structure models, the probability of generating an AER higher than that implied by the deterministic model ranges from a low of 72% to a high of 100% when the contractual interest rate is 6%. The high end of this range occurs under the term structure implied by the unrestricted model and because the simulations reveal a minimum random draw of 7.12% for the variance implied by the unrestricted term structure, which is greater than the long-term average of 6.89%. This occurrence can be traced to the parameter estimates used for the unrestricted model term structure simulations. The unrestricted model has a much smaller forecast variance than the other models.

In many cases, however, the probability of realizing an AER in excess of the deterministic rate when the contractual rate of interest is relatively high (10%), is still high. For example, the Merton, Vasicek, GBM, and CEV term structure models all suggest probabilities in excess of 50% that the simulated AER will exceed that estimated by the deterministic method even though the mean simulated value is less than the deterministic value. In short, there is a chance that the simulated AER will still eclipse the deterministic rate on average for these methods.

The remaining methods predict the opposite behavior. For example, at 6% and 8% contractual interest rates, the unrestricted term structure suggests a much higher than average chance that the simulated AER will exceed the deterministic rate. However, given a long-term rate of 6.89%, there is only a 9% probability of observing a simulated rate higher than the deterministic rate when the contractual rate is 10%. Presumably this occurs because of the intense downward pressure on a 10% interest rate given the long-term rate implied by the unrestricted term structure.

The last noteworthy comment on these results is with regard to truth-in-lending disclosure. As noted above, lenders typically disclose the APR, not the AER. The APRs associated with the deterministic AERs in Table 3 assuming 6%, 8%, and 10% rates of interest are 6.2391%, 8.2697%, and 10.3002%, respectively. Because of compounding, these APRs are lower than the AERs. Thus, the gap between what a lender would typically report and the potentially more accurate simulated annual effective rate of interest is even larger, with the results even more pronounced than discussed previously.

The Empirical Distribution of the Annual Effective Interest Rate <top>

To visually illustrate the distribution of AERs, Panels A­I of Figure 1 display graphs of the probability density functions of the AERs under each of the term structure models over the minimum and maximum range for a particular term structure. Further, these densities are for a contractual interest rate of 8%. As shown, varying shapes are possible with many of the term structure models predicting multi-modal distributions and exhibiting varying degrees of skewness and kurtosis.

In addition, for each distribution (with the exception of the unrestricted term structure), notice the truncation of the right tail of the distributions just beyond 12%. This is because the interest rate on the loan cannot exceed 12%. Recall, however, the AER is a compounded version of the actuarial rate which does include the impact of noninterest costs. Therefore, the AER can exceed 12% (as can the APR), but probably in aggregate does not exceed 12% very often (or by very much), as the graphs demonstrate. The unrestricted term structure model is a tighter distribution that never generates AERs beyond 12% because of the model’s strong mean-reverting tendencies (toward 6.89%).

Thus, successively high interest rates that could potentially put some weight in the right tail of the distribution do not occur as frequently under the unrestricted term structure model assumption as with the other models.

Figure 1. Probability Density Functions of the AERs Under Each of the Nine Term Structure Models

Panel A: Unrestricted Term Structure

Panel B: Merton Term Structure:

Panel C: Vasicek Term Structure

Panel D: CIR-SR Term Structure

Panel E: Dothan Term Structure

Panel F: GBM Term Structure

Panel G: BS Term Structure

Panel H: CIR-VR Term Structure

Panel I: CEV Term Structure

Conclusions <top>

This article has presented an exposition of some of the practical limitations of truth-in-lending type disclosure—namely, the potential unimportance of disclosing the annual percentage rate of interest. Without consensus on what noninterest costs to include, the rate disclosed will always be inaccurate and potentially useless for borrowers to use in comparing the cost of credit between two or more sources. Even if the issue surrounding which noninterest costs to include is standardized (and correctly so from an economic perspective), the annual effective rate of interest is a more accurate estimate to use than the annual percentage rate of interest. Repayments at intervals other than annually imply the annual percentage rate understates the cost of credit because of compounding.

Still, accurate cost of credit disclosure for the very common adjustable-rate mortgage remains elusive under present procedures, but could be improved using the methodology outlined here. The impact of accounting for a stochastic term structure cannot be overstated. The APRs that normally would get disclosed on a loan of the type outlined in the empirical section of this article are many basis points different than the simulated AERs. For example, for the loan outlined in this study with a contractual rate of interest of 8%, the deterministic APR is 8.27%, while the range of possible stochastic AERs is 7.62% to 9.96%.

A limitation to the present research is that the distribution of US Treasury securities (the index to which future interest rates are tied) must be known to apply the Monte Carlo approach. Nine term structures commonly used in empirical and analytical research are used in this study, but there is no guarantee any of these term structures are the correct one(s). Nonetheless, the method does provide flexibility, as other distributions and term structures still can be used.

More importantly, borrowers do tend to refinance adjustable-rate mortgages when conditions make it optimal to do so, and often prepay real estate mortgages. Through explicit modeling of the refinancing behavior of the borrower and the probability of prepayment, even more accurate estimates of the cost of credit could be obtained using only slight extensions to the modeling framework presented here.

References <top>

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Campbell, J., A. Lo, and A. MacKinlay. The Econometrics of Financial Markets. Princeton, NJ: Princeton University Press, 1997.

Chan, K., G. Karolyi, F. Longstaff, and A. Sanders. "An Empirical Comparison of Alternative Models of the Short-Term Interest Rate." J. Finance 47(1992): 1209­27.

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Financial Update staff. "Home-Equity Lending Takes Center Stage." Financial Update, Federal Reserve Bank of Atlanta, 10(July­September 1997):3.

Fischer, M., and G. Pederson. "Evaluating Annually Repriced Adjustable-Rate Mortgages." Agr. Fin. Rev. 51(1991): 106­17.

Heath, D., R. Jarrow, and A. Morton. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation." Econometrica 60(1992):77­105.

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Merton, R. "Theory of Rational Option Pricing." Bell J. Econ. Manage. Sci. 4(1973):141­83.

Nowman, K. "Gaussian Estimation of Single-Factor Continuous Time Models of the Term Structure of Interest Rates." J. Finance 52(1997):1695­1706.

Stanton, R. "A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk." J. Finance 52(1997):1973­2002.

Vasicek, O. "An Equilibrium Character-ization of the Term Structure." J. Fin. Econ. 5(1977):177­88.

 

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